Answer
See graph (red).
domain $(-\infty,\infty)$, range $[-3,\infty)$.
Work Step by Step
Step 1. To obtain the graph of $f(x)=2(x+1)^2-3$ from $y=x^2$, shift the curve 1 unit to the left, stretch vertically by a factor of 2, then shift 3 units down. See graph (red).
Step 2. We can find the domain $(-\infty,\infty)$ and the range $[-3,\infty)$.